We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more precise result, we show that if any hereditary graph class $\mathcal{G}$ admits $O(n^{1-\epsilon})$ separators, then for any fixed $\delta\in(0,\epsilon)$ every $n$-vertex graph in $\mathcal{G}$ is a subgraph of the strong product of a graph $H$ with bounded tree-depth and a complete graph of size $O(n^{1-\epsilon+\delta})$. This result holds with $\delta=0$ if we allow $H$ to have tree-depth $O(\log\log n)$. Moreover, using extensions of classical isoperimetric inequalties for grids graphs, we show the dependence on $\delta$ in our results and the above $\text{td}(H)\in O(\log\log n)$ bound are both best possible. We prove that $n$-vertex graphs of bounded treewidth are subgraphs of the product of a graph with tree-depth $t$ and a complete graph of size $O(n^{1/t})$, which is best possible. Finally, we investigate the conjecture that for any hereditary graph class $\mathcal{G}$ that admits $O(n^{1-\epsilon})$ separators, every $n$-vertex graph in $\mathcal{G}$ is a subgraph of the strong product of a graph $H$ with bounded tree-width and a complete graph of size $O(n^{1-\epsilon})$. We prove this for various classes $\mathcal{G}$ of interest.

翻译：我们研究了允许强次线性分割子的遗传图类的积结构。我们将此类图刻画为星图与大小呈强次线性完全图的强积的子图。在更精确的结果中，我们证明：若任一遗传图类$\mathcal{G}$允许$O(n^{1-\epsilon})$分割子，则对任意固定的$\delta\in(0,\epsilon)$，$\mathcal{G}$中每个$n$顶点图都是树深有界图$H$与大小为$O(n^{1-\epsilon+\delta})$的完全图之强积的子图。若允许$H$的树深为$O(\log\log n)$，则该结果在$\delta=0$时亦成立。此外，利用网格图经典等周不等式的推广，我们证明结果中关于$\delta$的依赖关系及上述$\text{td}(H)\in O(\log\log n)$界均为最优。我们证明树宽有界的$n$顶点图是树深为$t$的图与大小为$O(n^{1/t})$的完全图之积的子图，该结果亦为最优。最后，我们研究了如下猜想：对任一允许$O(n^{1-\epsilon})$分割子的遗传图类$\mathcal{G}$，$\mathcal{G}$中每个$n$顶点图都是树宽有界图$H$与大小为$O(n^{1-\epsilon})$的完全图之强积的子图。我们针对若干有意义的图类$\mathcal{G}$证明了该猜想。